Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
12
![**Problem Statement:**
Find the cost function \(C(x)\) if the marginal cost function is \(C'(x) = 3x - 4\) and the fixed cost is $8.
**Solution:**
To find the cost function \(C(x)\), we need to integrate the marginal cost function \(C'(x) = 3x - 4\).
1. **Integrate \(C'(x)\):**
\[
C(x) = \int (3x - 4) \, dx
\]
2. **Perform the integration:**
\[
C(x) = \int 3x \, dx - \int 4 \, dx
\]
This gives:
\[
C(x) = \frac{3x^2}{2} - 4x + C
\]
Where \(C\) is the constant of integration.
3. **Determine the constant of integration:**
Given the fixed cost is $8, we know that when \(x = 0\), \(C(x) = 8\).
Substitute \(x = 0\) into the cost function:
\[
8 = \frac{3(0)^2}{2} - 4(0) + C
\]
Therefore, \(C = 8\).
4. **Write the final cost function:**
\[
C(x) = \frac{3x^2}{2} - 4x + 8
\]
So, the cost function \(C(x)\) is \( \boxed{\frac{3x^2}{2} - 4x + 8} \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F02b4a1f4-9e35-42db-aa2a-674bb26df94b%2F5e7c58c1-7ecb-4d82-a089-9a8881615aa8%2Foi68tfg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the cost function \(C(x)\) if the marginal cost function is \(C'(x) = 3x - 4\) and the fixed cost is $8.
**Solution:**
To find the cost function \(C(x)\), we need to integrate the marginal cost function \(C'(x) = 3x - 4\).
1. **Integrate \(C'(x)\):**
\[
C(x) = \int (3x - 4) \, dx
\]
2. **Perform the integration:**
\[
C(x) = \int 3x \, dx - \int 4 \, dx
\]
This gives:
\[
C(x) = \frac{3x^2}{2} - 4x + C
\]
Where \(C\) is the constant of integration.
3. **Determine the constant of integration:**
Given the fixed cost is $8, we know that when \(x = 0\), \(C(x) = 8\).
Substitute \(x = 0\) into the cost function:
\[
8 = \frac{3(0)^2}{2} - 4(0) + C
\]
Therefore, \(C = 8\).
4. **Write the final cost function:**
\[
C(x) = \frac{3x^2}{2} - 4x + 8
\]
So, the cost function \(C(x)\) is \( \boxed{\frac{3x^2}{2} - 4x + 8} \).
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